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Abstract: Abstract In this paper, we study the regularization of 3D curves connecting two points. We propose an energy-based formulation which is an extension to 3D of the geodesic active contours introduced in 2D by Caselles et al. in 1997. By minimizing this energy we try to minimize the curve length but keeping the curve close to the original one. The energy depends on a regularization parameter which determines the smoothness of the regularized curve. We show the Euler-Lagrange equation of the proposed energy using the arc-length parameterization of the curve. We interpret the Euler-Lagrange equation in terms of the Frenet–Serret frame and we study some qualitative properties of the energy minima. We apply the steepest-descent method to approximate the local minima of the energy using an evolution equation. We propose a numerical scheme to solve the evolution equation and we present some experiments on real data in the context of aortic centerline regularization. PubDate: 2022-05-09

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Abstract: Abstract We focus on certain non-linear, non-convex, non-coercive systems of PDEs in three dimensions that are directly motivated by inverse problems in conductivity for the three-dimensional case. It turns out that such systems are variational, as they formally are the Euler–Lagrange systems associated with an explicit first-order functional, and thus we exploit both its variational structure as well as its connection to inverse problems. In particular, boundary conditions play a central role. PubDate: 2022-05-05

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Abstract: Abstract We provide a study of Blaschke–Santaló diagrams, i.e. complete systems of inequalities, for the inradius, diameter, and circumradius, measured with respect to different gauges. This contrasts previous works on those diagrams, which are all considered for the Euclidean measure. By proving several new inequalities and properties between these three functionals, we compute the intersection and the union over all possible gauges of those diagrams, showing that they coincide with the corresponding diagrams of a parallelotope and (in the planar case) a triangle, respectively. We also show that the planar spaces whose unit balls are regular pentagons or hexagons play an important role in the understanding of further extreme behaviours. PubDate: 2022-05-05

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Abstract: Abstract The k–generalized Pell sequence \(P^{(k)}:=(P_n^{(k)})_{n\ge -(k-2)}\) is the linear recurrence sequence of order k, whose first k terms are \(0, \ldots , 0,1\) and satisfies the relation \( P_{n}^{(k)} =2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}\) , for all \( n,k\ge 2 \) . In this paper, we investigate about integers that have at least two representations as a difference between a k–Pell number and a perfect power. In order to exhibit a solution method when b is known, we find all the integers c that have at least two representations of the form \( P_{n}^{(k)} - b^{m}\) for \( b\in [2,10]\) . This paper extends the previous works in Ddamulira et al. (Proc. Math. Sci. 127: 411–421, 2017) and Erazo et al. (J. Number Theory 203: 294–309, 2019). PubDate: 2022-04-29

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Abstract: Abstract Let M be a multimeasure defined on a \(\sigma \) -algebra and taking values in the family of bounded non-empty subsets of a Banach space X. We prove that M admits a control measure whenever X contains no subspace isomorphic to \(c_0(\omega _1)\) . The additional assumption on X is shown to be essential. PubDate: 2022-04-27

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Abstract: Abstract In this work, we examine the solvability of non-linear 2D Volterra integral equations through Petryshyn fixed point theorem in Banach space \(C([0, c]\times [0, d]).\) Our work covers many previous results that occur in non-linear analysis and some real-world problems under some weaker conditions. The efficiency of our work is presented by some examples. PubDate: 2022-04-24

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Abstract: Abstract In this article, we prove new discrete Picone inequalities, associated to non local elliptic operators as the fractional \( p- \) Laplace operator, denoted by \(\left( -\Delta \right) ^{s}_{p} u\) and defined as: $$\begin{aligned} \left( -\Delta \right) ^{s}_{p} u(x) := 2\, \mathbf{P.V. } \int _{\mathbb {R}^{N}}\dfrac{\left u(x) - u(y)\right ^{p-2}(u(x) - u(y))}{\left x-y\right ^{N + ps} }\,dy \end{aligned}$$ where \( p>1, \) \( 0< s < 1 \) and P.V. denotes the Cauchy principal value. These results lead to new applications as existence, non-existence and uniqueness of weak positive solutions to problems involving fractional and non homogeneous operators as \( (-\Delta )_{p}^{s_{1}} + (-\Delta )_{q}^{s_{2}}\) , where \( s_{1}, s_{2} \in (0,1) \) and \( 1< q, p<\infty \) . For this class of operators, we further obtain comparison principles, a Sturmian comparison principle and a Hardy-type inequality with weight. Finally, we also establish some qualitative results for nonlinear and non local elliptic systems with sub-homogeneous growth. PubDate: 2022-04-22

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Abstract: Abstract This paper introduces some inverse sequences of different polyhedra all based on finite approximations of a compact metric space so they can be used to capture the shape type of the original space. It is shown that they are HPol-expansions, proving the so-called general principle. We use these sequences to compute explicitly some inverse persistent homology groups of a space and measure its errors in the approximation process. PubDate: 2022-04-20

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Abstract: Abstract This paper investigates the solution of cobweb economic models when there is a Hilfer fractional derivative in the demand function or in the supply function. Particular cases when the Hilfer fractional derivative reduces to the Liouville–Caputo fractional derivative and the Riemann–Liouville fractional derivative are discussed. Moreover, the solution of the cobweb economic model with the Riemann–Liouville fractional derivative, which is derived as a special case of one of the main results in this paper, is also believed to be new. Two numerical examples, which interpret and further analyze the investigated cobweb theory, are presented to deliberate the illustrations and comparisons of the cobweb theory. Conclusions based on the graphical illustrations are outlined in detail. These illustrations highlight the advantage of the Hilfer fractional derivative over the aforementioned other two fractional derivatives, that is, the Liouville–Caputo fractional derivative and the Riemann–Liouville fractional derivative. PubDate: 2022-04-18

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Abstract: Abstract In this paper it is shown that for the ordinary Dirichlet series, \(\sum _{j=0}^{\infty }\frac{\alpha _{j}}{(j+1)^{s}}\) , \(\alpha _{0}=1\) , of a class, say \({\mathcal {P}}\) , that contains in particular the series that define the Riemann zeta and the Dirichlet eta functions, there exists \(\lim _{n\rightarrow \infty }\rho _{n}/n\) , where the \(\rho _{n}\) ’s are the Henry lower bounds of the partial sums of the given Dirichlet series, \(P_{n}(s)=\sum _{j=0}^{n-1}\frac{\alpha _{j}}{(j+1)^{s}}\) , \(n>2\) . Likewise it is given an estimate of the above limit. For the series of \({\mathcal {P}}\) having positive coefficients it is shown the existence of the \(\lim _{n\rightarrow \infty }a_{P_{n}(s)}/n\) , where the \(a_{P_{n}(s)}\) ’s are the lowest bounds of the real parts of the zeros of the partial sums. Furthermore it has been proved that \(\lim _{n\rightarrow \infty }a_{P_{n}(s)}/n=\lim _{n\rightarrow \infty }\rho _{n}/n\) . PubDate: 2022-04-15

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Abstract: Abstract In this paper, we characterize meromorphic solutions \(f(z_1,z_2),g(z_1,z_2)\) to the generalized Fermat Diophantine functional equations \(h(z_1,z_2)f^m+k(z_1,z_2)g^n=1\) in \({\mathbf {C}}^2\) for integers \(m,n\ge 2\) and nonzero meromorphic functions \(h(z_1,z_2),k(z_1,z_2)\) in \({\mathbf {C}}^2\) . Meromorphic solutions to associated partial differential equations are also studied. PubDate: 2022-04-12

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Abstract: Abstract We study the existence of solutions for some nonlinear elliptic boundary value problems, whose general form is: $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\mathrm{\;div}(M(x)\nabla u) + g(u) \nabla u ^2 = -\mathrm{\;div}(E(x,u)) + f(x) &{} \text{ in }\, \text{\O}mega , \\ u = 0 &{} \text{ on }\, \partial \text{\O}mega , \end{array}\right. } \end{aligned}$$ where \(\text{\O}mega \) is an open bounded subset of \(\mathbb {R}^N\) and \(f\in \,L^1(\text{\O}mega )\) . Despite this poor summability, we prove the existence of finite energy solutions due to the effect of the presence of the term \(g(u) \nabla u ^2 \) . PubDate: 2022-04-09

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Abstract: Abstract The classification problem for simple \({\mathfrak {sl}(2)}\) -modules leads in a natural way to the study of the category of finite rank torsion free \({\mathfrak {sl}(2)}\) -modules and its subcategory of rational \({\mathfrak {sl}(2)}\) -modules. We prove that the rationalization functor induces an identification between the isomorphism classes of simple modules of these categories. This raises the question of what is the precise relationship between other invariants associated with them. We give a complete solution to this problem for the Grothendieck and Picard groups, obtaining along the way several new results regarding these categories that are interesting in their own right. PubDate: 2022-04-02

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Abstract: Abstract We prove that, under a suitable rescaling of the integrable kernel defining the nonlocal diffusion terms, the corresponding sequence of solutions of the Shigesada–Kawasaki–Teramoto nonlocal cross-diffusion problem converges to a solution of the usual problem with local diffusion. In particular, the result may be regarded as a new proof of existence of solutions for the local diffusion problem. PubDate: 2022-04-02

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Abstract: Abstract It follows from a theorem of Rosenthal that a compact space is ccc if and only if its every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces \({\mathcal {C}}\) we define its orthogonal \({\mathcal {C}}^\perp \) as the class of all compact spaces for which every continuous image in \({\mathcal {C}}\) is metrizable. We study how this operation relates classes where centeredness is scarce with classes where it is abundant (like Eberlein and ccc compacta), and also classes where independence is scarce (most notably weakly Radon-Nikodým compacta) with classes where it is abundant. We study these problems for zero-dimensional compact spaces with the aid of Boolean algebras, and show the main difficulties that arise when we deal with general settings. Our main results are the constructions of several relevant examples. PubDate: 2022-03-31

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Abstract: Abstract Let \(\mathfrak {A}\) be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Lie n-derivation \(\Delta :\mathfrak {A}\rightarrow \mathfrak {A}\) is of the form \(\Delta (u)=zu+\delta (u),\) where \(z\in \mathcal {Z}(\mathfrak {A})\) and \(\delta :\mathfrak {A}\rightarrow \mathfrak {A}\) is a multiplicative Lie n-derivation. The main result is then applied to some classical examples of unital rings with nontrivial idempotents such as triangular rings, matrix rings, nest algebras, and algebras of bounded linear operators. PubDate: 2022-03-31

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Abstract: Abstract Let \({\mathcal {R}}\) be a 2-torsion free commutative ring with unity and X be a locally finite pre-ordered set. We prove in this paper that every Lie higher derivation on the incidence algebra \(I(X,{\mathcal {R}})\) is proper. PubDate: 2022-03-30

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Abstract: Abstract We introduce some measures of the dependence such as the strong mixing and uniform mixing coefficients in von Neumann algebras and then define the noncommutative strong and uniform mixing sequences. We establish some notable nonncommutative mixing inequalities such as Ibragimov inequality. Moreover, we extend the notion of mixingale sequence to the noncommutative content and demonstrate a noncommutative \(L_1\) and weak law of large numbers for uniformly integrable \(L_1\) -mixingale sequences. In addition, we investigate the noncommutative \(L_p\) -near-epoch dependence and provide some conditions under which a noncommtative \(L_p\) -near-epoch dependent sequence is a noncommutative \(L_p\) -mixingale. Finally, we introduce the concept of noncommutative \(L_p\) -approximability and show that in the setting of quantum (noncommutative) probability spaces, an \(L_r\) -bounded and \(L_0\) -approximable sequence is \(L_p\) -approximable for \(1\le p\) and \(2p< r<\infty \) . PubDate: 2022-03-25

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Abstract: Abstract There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts. PubDate: 2022-03-24

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Abstract: Abstract In the theory of approximation, linear operators play an important role. The exponential-type operators were introduced four decades ago, since then no new exponential-type operator was introduced by researchers, although several generalizations of existing exponential-type operators were proposed and studied. Very recently, the concept of semi-exponential operators was introduced and few semi-exponential operators were captured from the exponential-type operators. It is more difficult to obtain semi-exponential operators than the corresponding exponential-type operators. In this paper, we extend the studies and define semi-exponential Bernstein, semi-exponential Baskakov operators, semi-exponential Ismail–May operators related to \(2x^{3/2}\) or \(x^{3}\) . Furthermore, we present a new derivation for the semi-exponential Post–Widder operators. In some examples, open problems are indicated. PubDate: 2022-03-24